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The Necessary and Sufficient condition that an algebraic curve has an algebraic Involute is that the
Arc Length is a two-valued algebraic function of the coordinates of the extremities. Furthermore, this function
is a Root of a Quadratic Equation whose Coefficients are rational functions of 
and 
.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 195, 1959.